3.2545 \(\int \frac{(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^2}{605 \sqrt{5 x+3}}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (72060 x+173063)}{96800}-\frac{35451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^3)/(11*Sqrt[1
- 2*x]*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(173063 + 72060*x))/9680
0 - (35451*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[10])

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Rubi [A]  time = 0.192527, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^2}{605 \sqrt{5 x+3}}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (72060 x+173063)}{96800}-\frac{35451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^3)/(11*Sqrt[1
- 2*x]*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(173063 + 72060*x))/9680
0 - (35451*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[10])

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Rubi in Sympy [A]  time = 19.1701, size = 105, normalized size = 0.93 \[ - \frac{37 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{605 \sqrt{5 x + 3}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{270225 x}{2} + \frac{2595945}{8}\right )}{60500} - \frac{35451 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8000} + \frac{7 \left (3 x + 2\right )^{3}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

-37*sqrt(-2*x + 1)*(3*x + 2)**2/(605*sqrt(5*x + 3)) + sqrt(-2*x + 1)*sqrt(5*x +
3)*(270225*x/2 + 2595945/8)/60500 - 35451*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/1
1)/8000 + 7*(3*x + 2)**3/(11*sqrt(-2*x + 1)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.142448, size = 78, normalized size = 0.69 \[ \frac{10 \left (-392040 x^3-1992870 x^2+2323271 x+2026687\right )+4289571 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{968000 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]

[Out]

(10*(2026687 + 2323271*x - 1992870*x^2 - 392040*x^3) + 4289571*Sqrt[10 - 20*x]*S
qrt[3 + 5*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(968000*Sqrt[1 - 2*x]*Sqrt[3 + 5*
x])

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Maple [A]  time = 0.02, size = 137, normalized size = 1.2 \[ -{\frac{1}{-1936000+3872000\,x}\sqrt{1-2\,x} \left ( 42895710\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-7840800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4289571\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-39857400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-12868713\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(3/2),x)

[Out]

-1/1936000*(1-2*x)^(1/2)*(42895710*10^(1/2)*arcsin(20/11*x+1/11)*x^2-7840800*x^3
*(-10*x^2-x+3)^(1/2)+4289571*10^(1/2)*arcsin(20/11*x+1/11)*x-39857400*x^2*(-10*x
^2-x+3)^(1/2)-12868713*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1
/2)[phone]*(-10*x^2-x+3)^(1/2))/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49339, size = 101, normalized size = 0.89 \[ -\frac{81 \, x^{3}}{20 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1647 \, x^{2}}{80 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{35451}{16000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{2323271 \, x}{96800 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2026687}{96800 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

-81/20*x^3/sqrt(-10*x^2 - x + 3) - 1647/80*x^2/sqrt(-10*x^2 - x + 3) + 35451/160
00*sqrt(10)*arcsin(-20/11*x - 1/11) + 2323271/96800*x/sqrt(-10*x^2 - x + 3) + 20
26687/96800/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.230112, size = 115, normalized size = 1.02 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (392040 \, x^{3} + 1992870 \, x^{2} - 2323271 \, x - 2026687\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 4289571 \,{\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1936000 \,{\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

1/1936000*sqrt(10)*(2*sqrt(10)*(392040*x^3 + 1992870*x^2 - 2323271*x - 2026687)*
sqrt(5*x + 3)*sqrt(-2*x + 1) - 4289571*(10*x^2 + x - 3)*arctan(1/20*sqrt(10)*(20
*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(10*x^2 + x - 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{4}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Integral((3*x + 2)**4/((-2*x + 1)**(3/2)*(5*x + 3)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.252568, size = 177, normalized size = 1.57 \[ -\frac{35451}{8000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6534 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 197 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 21456431 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{12100000 \,{\left (2 \, x - 1\right )}} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{151250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{75625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

-35451/8000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/12100000*(6534*(12*
sqrt(5)*(5*x + 3) + 197*sqrt(5))*(5*x + 3) - 21456431*sqrt(5))*sqrt(5*x + 3)*sqr
t(-10*x + 5)/(2*x - 1) - 1/151250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/
sqrt(5*x + 3) + 2/75625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))